Description

Multiscale multifractal analysis (MMA) (Gierałtowski et al., 2012) is a time series analysis method, designed to describe scaling properties of fluctuations within the signal analyzed. The main result of this procedure is the so called Hurst surface h(q,s) , which is a dependence of the local Hurst exponent h (fluctuation scaling exponent) on the multifractal parameter q (Kantelhardt et al., 2002) and the scale of observation s (data window width).

MMA is a generalization of the standard multifractal detrended fluctuation analysis (MF-DFA) method (Kantelhardt et al., 2002) and adds the dependence on scale, yielding a broader analysis of the fluctuation properties as well as more universal and stable results. As a generalization, the method corresponds directly to earlier fractal signal analysis methods. On the Hurst surface h(q,s) , the results of the standard DFA (Peng et al., 1994) method are represented by one (or two) single points corresponding to the exponent a (or a1 and a2 ). The results of standard MF-DFA are represented by a single line: a cross section of h(q,s) at a constant s (in standard realization at s= 360).

The Hurst exponent larger than 0.5 is a measure of persistency of the signal (West et al., 2003; Schroeder, 1992) meaning how probable is it that, if the signal has just increased, the next sample will also be an increase of the signal value (and conversely for a decreasing signal). As usual in multifractal analysis, for a q>0 parameter the large fluctuations in the data are analyzed, while, for q<0, - the small fluctuations.

MMA eliminates the need for initial assumptions about the presumed time scale of the investigated problem. The new method is able to characterize monofractality or multifractality of the time series in a wide range of frequencies (scales) simultaneously and may be easily applied to data with strong crossover properties. It can correctly recognize fractal properties of time series even for relatively small scales, and it is capable of properly recognizing different fractal properties in small and large scales simultaneously.

The Matlab code, MMA.m, in this directory contains function calculating Hurst surface h(q,s) for a time series.
Function name: MMA. As a parameter function takes name of a single file or directory with files to be analyzed. File needs to have .txt extension and contain at least 1200 datapoints.

Usage

Usage example:

MMA(filename) or MMA(directory_name)

where filename (e.g., C:\directory\filename.txt) should be a full-path filename; and directory_name (e.g., C:\directory) is the name of the directory containing the input files.

Options

The MMA.m file contains a set of parameters, which if needed, can be adjusted by the user:

overlapping = 0

0 - time series is partitioned into no overlapping windows of analysis,

1 - time series is partitioned into overlapping windows of analysis, step between consecutive windows is = 1 (much longer calculations)

Sample3_multifractal_Cauchy_distribution_noise - Random number series with Cauchy distribution (heavy-tailed distribution). Sample of a time series with strong multifractality stemming only from distribution.

Sample4_mono_multi_fractal_phase_transition - Special example of time series with strong multifractal properties for short scales s and with strong monofractal properties for small scales s. Time series used as an example in (Gierałtowski et al. 2012) (see Fig. 3).

Sample5_RR_interval_series_healthy - night-time RR interval series for a healthy, 25 years old man. Characteristic, highly repetitive shape of Hurst surface for RR interval series of healthy adults during sleep.

The corresponding output from MMA using the above input files can be found in the subfolder Sample_time_series_with_results/MMA_results/.